# Self-Expanders to Inverse Curvature Flows by Homogeneous Functions

## TL;DR

This paper investigates self-expanding solutions to a broad class of inverse curvature flows in Euclidean spaces, revealing symmetry properties and existence of specific self-expanders, extending previous results to more general flows.

## Contribution

It characterizes compact and non-compact self-expanders for inverse curvature flows with homogeneous functions, generalizing earlier inverse mean curvature flow results.

## Key findings

- Only round spheres are compact self-expanders.
- Complete non-compact self-expanders with cylindrical ends are rotationally symmetric.
- Existence of rotationally symmetric self-expanders asymptotic to cylinders with different radii.

## Abstract

In this paper, we study self-expanding solutions to a large class of parabolic inverse curvature flows by homogeneous symmetric functions of principal curvatures in Euclidean spaces. These flows include the inverse mean curvature flow and many nonlinear flows in the literature. We first show that the only compact self-expanders to any of these flows are round spheres. Secondly, we show that complete non-compact self-expanders to any of these flows with asymptotically cylindrical ends must be rotationally symmetric. Thirdly, we show that when such a flow is uniformly parabolic, there exist complete rotationally symmetric self-expanders which are asymptotic to two round cylinders with different radii. These extend some earlier results of inverse mean curvature flow to a wider class of flows.